Abstract
In this paper, we consider a nonlinear inverse problem occurring in nuclear science. Gamma rays randomly hit a semiconductor detector which produces an impulse response of electric current. Because the sampling period of the measured current is larger than the mean inter arrival time of photons, the impulse responses associated to different gamma rays can overlap: this phenomenon is known as pileup. In this work, it is assumed that the impulse response is an exponentially decaying function. We propose a novel method to infer the distribution of gamma photon energies from the indirect measurements obtained from the detector. This technique is based on a formula linking the characteristic function of the photon density to a function involving the characteristic function and its derivative of the observations. We establish that our estimator converges to the mark density in uniform norm at a logarithmic rate. A limited Monte-Carlo experiment is provided to support our findings.
Highlights
In this paper, we consider a nonlinear inverse problem arising in nuclear science: neutron transport or gamma spectroscopy
The electrodes generate an electric current called impulse response whenever the detector is hit by a particle, with an amplitude corresponding to the transferred energy
The shot-noise process in nuclear applications corresponding to the electric current is discretely observed for three minutes at a sampling rate of 10Mhz and the mean number of arrivals between two observations lies between 10 and 100
Summary
We consider a nonlinear inverse problem arising in nuclear science: neutron transport or gamma spectroscopy. ([24]) provide consistent and asymptotically normal estimators for parametric shot-noise processes with specific impulse responses In this contribution, we consider the particular case given by the following assumption. In recent works, [3, Brockwell, Schlemm] exploit the integrated version of (6) to recover the Lévy process L and show that the increments of L can be represented as: nh These quantities are only well estimated for high frequency observations so that we cannot rely on this method in our regular sampling scheme.
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